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Thursday, April 4, 2013

Oh, hey

As has become a sort of tradition by now, I've noticed the state of abandonment of the blog once again and decided to revive it. Mostly, I blame the fact that I haven't been writing a lot, lately. Also, my last computer died in early January and I only got a new one a couple of weeks ago. Lost with that computer, incidentally, is all or most of my latest story, which was actually coming along nicely before everything went kablooey. 'Nicely' to be considered relative to my usual standards of creative output, that is.

While I muster the will to rewrite it or wait for a miracle to bring it back from the depths of a busted hard drive, here's a story from math class.

* * *

We are discussing how a particular mathematical transformation (a linear map) affects a rhombus. The computer shows first a parallelogram, then what appears to be a straight line. "Of course, this is actually a very thin parallelogram, not a line", says the professor."What would it imply, if it was an actual straight line?", she asks, while she tries to use the zoom tool to demonstrate.

The answer comes to mind, even before she has finished asking: "the matrix would not be invertible". Right behind me, another student says the same thing. The professor confirms this is so. Meanwhile, I am thinking, furiously, "How the fuck did I know that?"

It's not that it's an odd thing, or an unfamiliar subject. A few moments later, I figured out a couple of satisfactory answers as to why it must be so. It's just that, the moment I knew it, I had not gone through any of the intermediate steps in my mind. The answer seemed obvious in itself, as though a cached thought. And yet, as far as I recall, I have not come across this particular kind of question before. Did I do so and forget? Or did I just happen to have a cool flash of mathematical intuition?

Mathematical details, for those so inclined: A linear map is, for our purposes, a function that given a vector returns another vector, with some properties. Namely, if X and Y are vectors and f is a linear map, then f(X+Y)=f(X)+f(Y), and if k is a real number, then f(k*X)=k*f(X). We'll be working with vectors in R2, which can be thought of as arrows in a plane. Every linear map has an associated matrix, and iff the map has an inverse map (i.e, a map g such that g(f(X))=X for all vectors X) then the associated matrix is invertible (i.e., there is another matrix which, multiplied by it, returns the identity matrix). If you don't know what a vector or a matrix are, I suggest Wikipedia, an algebra class, or giving up. The idea above of applying the map to a rhombus means, essentially, that you apply the map to the vectors of every point of the rhombus and see what happens. In practice, all you actually need to do is apply it to the vertices of the rhombus and then connect the dots.

Now, the reasons a rhombus couldn't be turned into a line by an invertible matrix can be expressed in many ways. The first, is that it would imply that the transforms of all the corners of the rhombus are in a straight line. Because of the second property above, all vectors in the same straight line (which are all multiples of each other) also go to a straight line after mapping. If two separate lines in the rhombus go to the same line in its map, that implies there are different vectors that are being mapped to the same vector. A function that takes two inputs and sends them to the same output cannot have an inverse function. Therefore, the matrix associated with the map cannot be invertible.

Another way of looking at it is that if the map turns a two-dimensional figure such as a rhombus into a one-dimensional line, there must be some direction in space it makes zero. (This is by no means rigorous proof, just the sort of thing that would be stored in my brain and influence my intuitions). A linear map always maps zero to zero, so if there is another vector that also goes to zero, it cannot have an inverse.

A third way, which came to mind on the bus ride home, is that the determinant of the transformation of a matrix tells you how it changes areas after transforming. (This is only obvious to me because it's an important part of the change of variable theorem in integration. Being a physics student, calculus is something I use much more often than algebra). A straight line has area zero, which means the determinant of the transformation is also zero, and there's a theorem that states that if the determinant is zero, the matrix is not invertible.